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Mirrors > Home > MPE Home > Th. List > distgp | Structured version Visualization version GIF version |
Description: Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
distgp.1 | ⊢ 𝐵 = (Base‘𝐺) |
distgp.2 | ⊢ 𝐽 = (TopOpen‘𝐺) |
Ref | Expression |
---|---|
distgp | ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ Grp) | |
2 | simpr 486 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 = 𝒫 𝐵) | |
3 | distgp.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 3 | fvexi 6854 | . . . . 5 ⊢ 𝐵 ∈ V |
5 | distopon 22299 | . . . . 5 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ (TopOn‘𝐵)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ 𝒫 𝐵 ∈ (TopOn‘𝐵) |
7 | 2, 6 | eqeltrdi 2847 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 ∈ (TopOn‘𝐵)) |
8 | distgp.2 | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐺) | |
9 | 3, 8 | istps 22235 | . . 3 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵)) |
10 | 7, 9 | sylibr 233 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopSp) |
11 | eqid 2738 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
12 | 3, 11 | grpsubf 18785 | . . . . 5 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
13 | 12 | adantr 482 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
14 | 4, 4 | xpex 7680 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V |
15 | 4, 14 | elmap 8768 | . . . 4 ⊢ ((-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵)) ↔ (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
16 | 13, 15 | sylibr 233 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵))) |
17 | 2, 2 | oveq12d 7370 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = (𝒫 𝐵 ×t 𝒫 𝐵)) |
18 | txdis 22935 | . . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵)) | |
19 | 4, 4, 18 | mp2an 691 | . . . . . 6 ⊢ (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵) |
20 | 17, 19 | eqtrdi 2794 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = 𝒫 (𝐵 × 𝐵)) |
21 | 20 | oveq1d 7367 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝒫 (𝐵 × 𝐵) Cn 𝐽)) |
22 | cndis 22594 | . . . . 5 ⊢ (((𝐵 × 𝐵) ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) | |
23 | 14, 7, 22 | sylancr 588 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) |
24 | 21, 23 | eqtrd 2778 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) |
25 | 16, 24 | eleqtrrd 2842 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
26 | 8, 11 | istgp2 23394 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
27 | 1, 10, 25, 26 | syl3anbrc 1344 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 𝒫 cpw 4559 × cxp 5630 ⟶wf 6490 ‘cfv 6494 (class class class)co 7352 ↑m cmap 8724 Basecbs 17043 TopOpenctopn 17263 Grpcgrp 18708 -gcsg 18710 TopOnctopon 22211 TopSpctps 22233 Cn ccn 22527 ×t ctx 22863 TopGrpctgp 23374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7914 df-2nd 7915 df-map 8726 df-0g 17283 df-topgen 17285 df-plusf 18456 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-grp 18711 df-minusg 18712 df-sbg 18713 df-top 22195 df-topon 22212 df-topsp 22234 df-bases 22248 df-cn 22530 df-cnp 22531 df-tx 22865 df-tmd 23375 df-tgp 23376 |
This theorem is referenced by: (None) |
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